Fractional multiphase hereditary materials: Mellin Transforms and Multi-Scale Fractances

The rheological features of several complex organic natural tissues such as bones, muscles as well as of complex artificial polymers are well described by power-laws. Indeed, it is well-established that the time-dependence of the stress and the strain in relaxation/creep test may be well captured by power-laws with exponent β ∈ [0, 1]. In this context a generalization of linear springs and linear dashpots has been introduced in scientific literature in terms of a mechanical device dubbed spring-pot. Recently the authors introduced a mechanical analogue to spring-pot built upon a proper arrangements of springs and dashpots that results in Elasto-Viscous (EV) materials, as β ∈ [0, 1/2] and Visco-Elastic ones, as β ∈ [1/2, 1]. In this paper the authors will discuss the rheological description of the presence of multiple material phases that is highlighted by a linear combination of power-laws in the relaxation function G(t) with different exponents. Such rehological model is represented by a linear combination of fractional derivatives with different order and the inverse relations have been formulated in terms of the complex method Mellin transform. Additionally an alternative representation of direct and inverse relations of multi-phase fractional hereditary materials based on the exact mechanical description of spring-pot element will be discussed in the course of the paper.